The staggered algorithm remains a popular choice for solving multi-field coupled problems due to its stability and ease of implementation. However, its iterative nature can lead to slow convergence and increased computational expense in strongly coupled scenarios. This talk introduces the concept of the degree of coupling (DOC), a key parameter derived from the block structure of the Jacobian matrix that governs the convergence behavior of staggered iterations. We show that the staggered scheme achieves first-order convergence, where the DOC determines the reduction factor of both residual and error. When the DOC approaches zero, rapid convergence is observed; when it approaches unity, convergence becomes sluggish; and when it exceeds one, convergence is no longer guaranteed. In phase-field fracture modeling, for example, the DOC tends to increase with loading and eventually oscillates just below one, resulting in numerous iterations despite eventual convergence. To address this limitation, we propose a Schur complement-enhanced Newton iterative computing (SCENIC) that incorporates inter-field coupling directly into the update of individual fields. This approach dramatically reduces both iteration count and computational time. Numerical experiments on phase field fracture benchmarks and a coupled thermo-mechanical fracture problem demonstrate that the proposed method outperforms conventional staggered and L-BFGS schemes in terms of efficiency. REFERENCES [1] Y. Zhao, J. Jiang, B. Lyu and Y. Shen. Staggered algorithms for coupled problems: Convergence analysis and application to phase field modeling of thermal cracking. Computer Methods in Applied Mechanics and Engineering, Vol. 446, 118238, 2025. [2] J. Jiang, Y. Zhao, B. Lyu and Y. Shen. Schur-complement-enhanced iterative solvers for coupled problems and phase field fracture models. Computer Methods in Applied Mechanics and Engineering, Vol. 449, 118509, 2026.